Parallel group topologies on Prüfer groups

Question

Let $p$ be a prome number. Are there group topologies $\mathcal S$ and $\mathcal T$ on $\Bbb Z_{p^\infty}$ such that $$\mathcal T \nsubseteq \mathcal S,~~\mathcal S \nsubseteq \mathcal T$$

Answer 1

I found a result in

Cardinal invariants and independence results in the poset of precompact group topologies, Alessandro Berarducci, Dikran Dikranjan, Marco Forti, Stephen Watson:

Let $G$ be an infinite abelian group. The power set of the power set of $G$ can be embedded in the poset of all Hausdorff group topologies on $G$.

So there are $A,B\in \mathcal P(\mathcal P(\Bbb Z_{p^\infty}))$ with $$A\nsubseteq B,~~ B\nsubseteq A$$ $A$ and $B$ correspond to group topologies $\mathcal T$ and $\mathcal S$ with

$$\mathcal T \nsubseteq \mathcal S,~~\mathcal S \nsubseteq \mathcal T$$

But I do not know what $\mathcal T$ and $\mathcal S$ are.